A series associated to Rankin-Selberg L -function and modified K -Bessel function

Zagier, in 1981, conjectured that the constant term of an automorphic function associated to the Ramanujan delta function, i.e. y12Σ ∞ n=1 τ2(n)e-4πny, has a connection with the nontrivial zeros of ζ(s). This conjecture was finally proved by Hafner and Stopple in 2000. Recently, Chakraborty et al. extended this observation for any normalized Hecke eigenform over SL2(Z). In this paper, we study the infinite series Σ ∞ n=1 cℓ f (n)nν/2Kν (y √ n) for ℓ = 1, 2, where cf (n) denotes the nth Fourier coefficient of a normalized Hecke eigenform f(z) and Kν represents the modified Bessel function of the second kind of order ν. We generalize a recent identity of Berndt et al. We also observe that the aforementioned series corresponding to ℓ = 2 has a connection with the nontrivial zeros of ζ(s).